Prime numbers, prime factors and composite numbers

In a nutshell

Prime and composite numbers are special categories of numbers based on how many factors they have. There are $$25$$ prime numbers and $$74$$ composite numbers between $$1$$ and $$100$$.  

Prime numbers and composite numbers

Prime numbers only have two factors, $$1$$ and itself. Composite numbers have more than two factors. 

Example 1

Identify whether the following numbers are prime or composite: $$2,10,13,25$$.

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Note: ​$$1$$ is neither a prime or composite number because it only has one factor, $$1$$. 

Prime factors

Factors of a number which are prime are prime factors. They can be found using a factor tree to breakdown a number into its prime factors which can be multiplied together to give the original number. This can be done by following the procedure below: 

PROCEDURE

Example 2

What is $$32$$ as a product of its prime factors?

Write $$32$$​ on the top, and find two numbers that multiply to give $$32$$:

​$$32=2\times16$$

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Write $$2$$ and $$16$$ beneath $$32$$. $$2$$​ is a prime number so circle it.

$${\begin{aligned} &\space \space \,\,\,\,32\\\ &\,\,\swarrow\searrow& \\ &\textcircled{2} \quad16 \end{aligned}}$$​​

As $$16$$ is not a prime number, find two numbers that multiply to give $$16$$ such as: $$2 \times 8$$​. Add this to the factor tree and circle $$2$$ since it is prime. ​

$$\quad{\begin{aligned} &\space \space \,\,\,\,32\\\ &\,\,\swarrow\searrow& \\ &\textcircled{2} \quad16 \\ &\quad\,\,\swarrow\searrow\\ &\space\space\space\,\,\textcircled2 \,\,\quad8 \end{aligned}}$$

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As $$8$$ is not a prime number, find two numbers that multiply to give $$8$$ such as: $$2 \times 4$$​. Add this to the factor tree and circle $$2$$ since it is prime.

$$\quad\quad{\begin{aligned} &\space \space \,\,\,\,32\\\ &\,\,\swarrow\searrow& \\ &\textcircled{2} \quad16 \\ &\quad\,\,\swarrow\searrow\\ &\space\space\space\,\,\textcircled2 \,\,\quad8 \\ &\quad\quad\,\,\,\swarrow\searrow \\ &\quad\space\space\,\,\textcircled2 \quad\space\,\,\, 4 \end{aligned}}$$​​

​As $$4$$​ is not a prime number, find two numbers that multiply to give $$4$$​ such as $$2 \times 2$$​. Add this to the factor tree and circle both $$2$$s since they are prime.

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​$$\quad\quad\quad\space{\begin{aligned} &\space \space \,\,\,\,32\\\ &\,\,\swarrow\searrow& \\ &\textcircled{2} \quad16 \\ &\quad\,\,\swarrow\searrow\\ &\space\space\space\,\,\textcircled2 \,\,\quad8 \\ &\quad\quad\,\,\,\swarrow\searrow \\ &\quad\space\space\,\,\textcircled2 \quad\space\,\,\, 4 \\ &\quad\quad\quad\space\,\,\swarrow\searrow \\ &\quad\quad\quad\textcircled2 \quad\space \,\,\textcircled2\end{aligned}}$$​​

Only prime factors remain so the tree is complete. 

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Write $$32$$ as a product of its prime factors.​

Therefore, $$\underline{2 \times 2\times 2\times 2\times 2=32.}$$

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Note: ​$$1$$ is not a prime number so it does not appear in the factor tree.